From adb7321b939dfc623c8ba032b2a4c89f8735a851 Mon Sep 17 00:00:00 2001 From: Andreas Tsouchlos Date: Thu, 13 Jun 2024 17:41:59 +0200 Subject: [PATCH] Add changes made before submission for review --- letter.tex | 186 +++++++++++++++++++++++++++++++---------------------- 1 file changed, 109 insertions(+), 77 deletions(-) diff --git a/letter.tex b/letter.tex index a651373..536fcbf 100644 --- a/letter.tex +++ b/letter.tex @@ -56,7 +56,7 @@ \pgfplotsset{colorscheme/cel} \newcommand{\figwidth}{\columnwidth} -\newcommand{\figheight}{0.75\columnwidth} +\newcommand{\figheight}{0.7\columnwidth} \pgfplotsset{ FERPlot/.style={ @@ -107,17 +107,17 @@ \begin{abstract} -In this paper, the proximal decoding algorithm is considered within the +In this paper, the proximal decoding algorithm described in, e.g., \cite{proximal_paper}, is considered within the context of \textit{additive white Gaussian noise} (AWGN) channels. An analysis of the convergence behavior of the algorithm shows that proximal decoding inherently enters an oscillating behavior of the estimate after a certain number of iterations. Due to this oscillation, frame errors arising during decoding can often -be attributed to only a few remaining wrongly decoded bits. +be attributed to only a few remaining wrongly decoded bit positions. In this letter, an improvement of the proximal decoding algorithm is proposed -by appending an additional step, in which these erroneous components are +by establishing an additional step, in which these erroneous positions are attempted to be corrected. -We suggesst an empirical rule with which the components most likely needing +We suggest an empirical rule with which the components most likely needing correction can be determined. Using this insight and performing a subsequent ``ML-in-the-list'' decoding, a gain of up to 1 dB is achieved compared to conventional @@ -151,10 +151,12 @@ While the established decoders for LDPC codes, such as belief propagation (BP) and the min-sum algorithm, offer good decoding performance, they are generally not optimal and exhibit an error floor for high \textit{signal-to-noise ratios} (SNRs) \cite{channel_codes_book}, making them -unsuitable for applications with extreme reliability requirements. +inadequate for applications with extreme reliability requirements. Optimization based decoding algorithms are an entirely different way of -approaching the decoding problem. +approaching the decoding problem; +they map the decoding problem onto an optimization problem in order to +leverage the vast knowledge from the field of optimization theory. A number of different such algorithms have been introduced. The field of \textit{linear programming} (LP) decoding \cite{feldman_paper}, for example, represents one class of such algorithms, based on a relaxation @@ -167,26 +169,26 @@ Proximal decoding relies on a non-convex optimization formulation of the \textit{maximum a posteriori} (MAP) decoding problem. The aim of this work is to improve upon the performance of proximal decoding by -first presenting an examination of the algorithm's behavior and then suggesting +first presenting an analysis of the algorithm's behavior and then suggesting an approach to mitigate some of its flaws. This analysis is performed for \textit{additive white Gaussian noise} (AWGN) channels. We first observe that the algorithm initially moves the estimate in -the right direction, however, in the final steps of the decoding process, +the right direction; however, in the final steps of the decoding process, convergence to the correct codeword is often not achieved. -Furthermore, we suggest that the reason for this behavior is the nature +Subsequently, we attributed this behavior to the nature of the decoding algorithm itself, comprising two separate gradient descent steps working adversarially. -We propose a method mitigate this effect by appending an -additional step to the decoding process. +We, thus, propose a method to mitigate this effect by appending an +additional step to the iterative decoding process. In this additional step, the components of the estimate with the highest probability of being erroneous are identified. New codewords are then generated, over which an ``ML-in-the-list'' \cite{ml_in_the_list} decoding is performed. A process to conduct this identification is proposed in this paper. Using the improved algorithm, a gain of up to -1 dB can be achieved compared to conventional proximal decoding, +$\SI{1}{dB}$ can be achieved compared to conventional proximal decoding, depending on the decoder parameters and the code. @@ -200,7 +202,7 @@ When considering binary linear codes, data words are mapped onto codewords, the lengths of which are denoted by $k \in \mathbb{N}$ and $n \in \mathbb{N}$, respectively, with $k \le n$. The set of codewords $\mathcal{C} \subset \mathbb{F}_2^n$ of a binary linear -code can be represented using the parity-check matrix +code can be characterized using the parity-check matrix $\boldsymbol{H} \in \mathbb{F}_2^{m \times n} $, where $m$ represents the number of parity-checks: % @@ -230,7 +232,7 @@ estimate of the transmitted codeword, denoted as $\hat{\boldsymbol{c}} \in \mathbb{F}_2^n$. A distinction is made between $\boldsymbol{x} \in \left\{\pm 1\right\}^n$ and $\tilde{\boldsymbol{x}} \in \mathbb{R}^n$, -the former denoting the BPSK symbol physically transmitted over the channel and +the former denoting the BPSK symbols transmitted over the channel and the latter being used as a variable during the optimization process. The posterior probability of having transmitted $\boldsymbol{x}$ when receiving $\boldsymbol{y}$ is expressed as a \textit{probability mass function} (PMF) @@ -267,8 +269,8 @@ One such expression, formulated under the assumption of BPSK, is the .\end{align*}% % Its intent is to penalize vectors far from a codeword. -It comprises two terms: one representing the bipolar constraint -and one representing the parity constraint, incorporating all of the +It comprises two terms: one representing the bipolar constraint due to transmitting BPSK +and one representing the parity constraint, incorporating all information regarding the code. The channel model can be considered using the negative log-likelihood @@ -279,7 +281,7 @@ The channel model can be considered using the negative log-likelihood \boldsymbol{y} \mid \tilde{\boldsymbol{x}} \mright) \mright) .\end{align*} % -The information about the channel and the code are consolidated in the objective +Then, the information about the channel and the code are consolidated in the objective function \cite{proximal_paper} % \begin{align*} @@ -305,17 +307,17 @@ introduced, describing the result of each of the two steps: .\end{alignat} % An equation for determining $\nabla h(\boldsymbol{r})$ is given in -\cite{proximal_paper}. +\cite{proximal_paper}, where it is also proposed to initialized $\boldsymbol{s}=\boldsymbol{0}$. It should be noted that the variables $\boldsymbol{r}$ and $\boldsymbol{s}$ represent $\tilde{\boldsymbol{x}}$ during different stages of the decoding process. As the gradient of the code-constraint polynomial can attain very large values -in some cases, an additional step is introduced to ensure numerical stability: -every current estimate $\boldsymbol{s}$ is projected onto +in some cases, an additional step is introduced in \cite{proximal_paper} to ensure numerical stability: +every estimate $\boldsymbol{s}$ is projected onto $\left[-\eta, \eta\right]^n$ by a projection $\Pi_\eta : \mathbb{R}^n \rightarrow \left[-\eta, \eta\right]^n$, where $\eta$ -is a positive constant slightly larger than one, e.g., $\eta = 1.5$. +is a positive constant larger than one, e.g., $\eta = 1.5$. The resulting decoding process as described in \cite{proximal_paper} is presented in Algorithm \ref{alg:proximal_decoding}. @@ -346,13 +348,12 @@ presented in Algorithm \ref{alg:proximal_decoding}. \subsection{Analysis of the Convergence Behavior} In Fig. \ref{fig:fer vs ber}, the \textit{frame error rate} (FER), -\textit{bit error rate} (BER) and \textit{decoding failure rate} (DFR) of +\textit{bit error rate} (BER), and \textit{decoding failure rate} (DFR) of proximal decoding are shown for an LDPC code with $n=204$ and $k=102$ \cite[204.33.484]{mackay}. -A decoding failure is defined as a decoding operation returning an invalid -codeword, i.e., as non-convergence of the algorithm. +Hereby, a \emph{decoding failure} is defined as returning a \emph{non valid codeword}, i.e., as non-convergence of the algorithm. The parameters chosen for this simulation are $\gamma=0.05, \omega=0.05, -\eta=1.5$ and $K=200$. +\eta=1.5$ and $K=200$ ($K$ describing the maximum number of iterations). They were determined to offer the best performance in a preliminary examination, where the effect of changing multiple parameters was simulated over a wide range of values. @@ -367,7 +368,7 @@ the right direction. This would suggest that most frame errors occur due to only a few incorrectly decoded bits.% % -\begin{figure} +\begin{figure}[t] \centering @@ -417,14 +418,13 @@ decoded bits.% \end{figure}% % -An approach for lowering the FER might then be to append an ``ML-in-the-list'' +An approach for lowering the FER might then be to add an ``ML-in-the-list'' \cite{ml_in_the_list} step to the decoding process shown in Algorithm \ref{alg:proximal_decoding}. -This step consists in determining the $N \in \mathbb{N}$ most probable -erroneous bits, finding all variations of the current estimate with those bits -modified, and performing ML decoding on this list. +This step consists in determining the $N \in \mathbb{N}$ most probably +erroneous bit positions $\mathcal{I}'$, generating a list of $2^N$ codeword candidates out of the current estimate $\hat{\boldsymbol{c}}$ with bits in $\mathcal{I}'$ adopting all possible values, i.e., $\mathcal{L}'=\left\{ \hat{\boldsymbol{c}}'\in\mathbb{F}_2^n: \hat{c}'_i=\hat{c}_i, i\notin \mathcal{I}'\text{ and } \hat{c}'_i\in\mathbb{F}_2, i\in \mathcal{I}' \right\}$, and performing ML decoding on this list. -This approach crucially relies on identifying the most probable erroneous bits. +This approach crucially relies on identifying the most probably erroneous bits. Therefore, the convergence properties of proximal decoding are investigated. Considering (\ref{eq:s_update}) and (\ref{eq:r_update}), Fig. \ref{fig:grad} shows the two gradients along which the minimization is @@ -437,7 +437,7 @@ This behavior supports the conjecture that the reason for the high DFR is a failure to converge to the correct codeword in the final steps of the optimization process.% % -\begin{figure} +\begin{figure}[t] \centering \ifoverleaf @@ -538,7 +538,7 @@ optimization process.% $\nabla L\left(\boldsymbol{y} \mid \tilde{\boldsymbol{x}}\right)$ and $\nabla h \left( \tilde{\boldsymbol{x}} \right)$ for a repetition code with $n=2$. - Shown for $\boldsymbol{y} = \begin{bmatrix} -0.5 & 0.8 \end{bmatrix}$. + Shown for $\boldsymbol{y} = \begin{pmatrix} -0.5 & 0.8 \end{pmatrix}$. } \label{fig:grad} \end{figure}% @@ -548,7 +548,7 @@ In Fig. \ref{fig:prox:convergence_large_n}, we consider only component $\left(\tilde{\boldsymbol{x}}\right)_1$ of the estimate during a decoding operation for the LDPC code used also for Fig. 1. Two qualities may be observed. -First, we observe the average absolute values of the two gradients are equal, +First, we observe that the average absolute values of the two gradients are equal, however, they have opposing signs, leading to the aforementioned oscillation. Second, the gradient of the code constraint polynomial itself starts to @@ -605,16 +605,16 @@ oscillate after a certain number of iterations.% \subsection{Improvement Using ``ML-in-the-List'' Step} Considering the magnitude of the oscillation of the gradient of the code constraint -polynomial, some interesting behavior may be observed. -Fig. \ref{fig:p_error} shows the probability that a component of the estimate -is wrong, determined through a Monte Carlo simulation, when the components of -$\boldsymbol{c}$ are ordered from smallest to largest oscillation of -$\left(\nabla h\right)_i$. +polynomial, some interesting behavior may be observed. Let $\boldsymbol{i}'=(i'_1, \ldots, i_n')$ be a permutation of $\{1,\ldots, n\}$ such that $\left(\nabla h\right)_{i'}$ is arranged according to increasing variance of oscillation of its magnitude, i.e., $\text{Var}_\text{iter}(|\left(\nabla h\right)_{i'_1}|)\leq \cdots \leq \text{Var}_\text{iter}(|\left(\nabla h\right)_{i'_n}|)$ with $\text{Var}_\text{iter}(\cdot)$ denoting the empirical variance along the iterations. -The lower the magnitude of the oscillation, the higher the probability that the -corresponding bit was not decoded correctly. -This means that this magnitude is a suitable figure of merit for determining -the probability that a given component was decoded incorrectly.% +Hereafter, Fig. \ref{fig:p_error} shows Monte Carlo simulations of the probability that decoded bit $\hat{c}_i'$ at position $i'$ of the estimated codeword +is wrong. %, when the components of +%$\boldsymbol{c}$ are ordered from smallest to largest oscillation of +%$\left(\nabla h\right)_i$. +It can be observed that lower magnitudes of oscillation correlate with higher probability that the corresponding bit was not decoded correctly. +Thus, this magnitude might be used as a feasible indicator +%for determining the probability that a given component was decoded incorrectly and, thus, +for identifying erroneously decoded bit positions as $\mathcal{I}'=\{i_1', \ldots, i_N'\}$.% % \begin{figure}[H] \centering @@ -640,10 +640,9 @@ the probability that a given component was decoded incorrectly.% \fi \caption{Probability that a component of the estimated codeword - $\hat{\boldsymbol{c}}\in \mathbb{F}_2^n$ is erroneous for a (3,6) regular + $\boldsymbol{\hat{c}}\in \mathbb{F}_2^n$ is erroneous for a (3,6) regular LDPC code with $n=204, k=102$ \cite[\text{204.33.484}]{mackay}. - The indices $i'$ are ordered such that the amplitude of oscillation of - $\left(\nabla h\right)_{i'}$ increases with $i'$. + Indices $i'$ are ordered such that $|\left(\nabla h\right)_{i'_1}|\leq \cdots \leq |\left(\nabla h\right)_{i'_n}|$. Parameters used for the simulation: $\gamma = 0.05, \omega = 0.05, \eta = 1.5, E_b/N_0 = \SI{4}{dB}$. Simulated with $\SI{100000000}{}$ iterations using the all-zeros codeword.} @@ -656,25 +655,10 @@ If a valid codeword has been reached, i.e., if the algorithm has converged, we return this solution. Otherwise, $N \in \mathbb{N}$ components are selected based on the criterion presented above. -Beginning with the recent estimate $\hat{\boldsymbol{c}} \in \mathbb{F}_2^n$, -all variations of words with the selected components modified are then +Originating from $\boldsymbol{\hat{c}} \in \mathbb{F}_2^n$ resulting from proximal decoding, +the list $\mathcal{L}'$ of codeword candidates with bits in $\mathcal{I}'$ modified is generated and an ``ML-in-the-list'' step is performed. -\begin{algorithm} - \caption{ML-in-the-List algorithm.} - \label{alg:ml-in-the-list} - - \begin{algorithmic} - \STATE Find valid codewords under $\left(\hat{\boldsymbol{c}}_{l}\right)_{1=1}^{2^N}$ - \STATE \textbf{if} no valid codewords exist - \STATE \hspace{5mm} Compute $\langle \hat{\boldsymbol{c}}_l, \hat{\boldsymbol{c}} \rangle$ for all variations $\boldsymbol{c}_l$ - \STATE \textbf{else} - \STATE \hspace{5mm} Compute $\langle \hat{\boldsymbol{c}}_l, \hat{\boldsymbol{c}} \rangle$ for valid codewords - \STATE \textbf{end if} - \STATE \textbf{return} $\hat{\boldsymbol{c}}_l$ with highest $\langle \hat{\boldsymbol{c}}_l, \hat{\boldsymbol{c}} \rangle$ - \end{algorithmic} -\end{algorithm}% -% \begin{algorithm} \caption{Improved proximal decoding algorithm. } @@ -685,24 +669,62 @@ generated and an ``ML-in-the-list'' step is performed. \STATE \textbf{for} $K$ iterations \textbf{do} \STATE \hspace{5mm} $\boldsymbol{r} \leftarrow \boldsymbol{s} - \omega \left( \boldsymbol{s} - \boldsymbol{y} \right) $ \STATE \hspace{5mm} $\boldsymbol{s} \leftarrow \Pi_\eta \left(\boldsymbol{r} - \gamma \nabla h\left( \boldsymbol{r} \right) \right)$ - \STATE \hspace{5mm} $\boldsymbol{\hat{c}} \leftarrow \mathds{1} \left\{ \text{sign}\left( \boldsymbol{s} \right) = -1 \right\}$ - \STATE \hspace{10mm} \textbf{if} $\boldsymbol{H}\boldsymbol{\hat{c}} = \boldsymbol{0}$ \textbf{do} + \STATE \hspace{5mm} $\boldsymbol{\hat{c}} \leftarrow \mathds{1}_{ \left\{ \boldsymbol{s} \leq 0 \right\}}$ + \STATE \hspace{5mm} \textbf{if} $\boldsymbol{H}\boldsymbol{\hat{c}} = \boldsymbol{0}$ \textbf{do} \STATE \hspace{10mm} \textbf{return} $\boldsymbol{\hat{c}}$ \STATE \hspace{5mm} \textbf{end if} \STATE \textbf{end for} - \STATE $\textcolor{KITblue}{\text{Estimate $N$ wrong bit indices $\mathcal{I} = \{i_1,\ldots,i_N\}$}}$ - \STATE $\textcolor{KITblue}{\text{Generate candidate list $\left(\hat{\boldsymbol{c}}_{l}\right)_{l=1}^{2^N}$ by varying bits in $\mathcal{I}$}}$\vspace{1mm} - \STATE $\textcolor{KITblue}{\textbf{return ml\textunderscore in\textunderscore the\textunderscore list}\left(\left(\hat{\boldsymbol{c}}_l\right)_{1=1}^{2^N}\right)}$ + \STATE $\textcolor{KITblue}{\text{$\mathcal{I}'\leftarrow \{i_1',\ldots, i_N'\}$ (indices of $N$ probably wrong bits) + %$\mathcal{I} = \{i_1,\ldots,i_N\}$ + } + }$ + + \STATE $\textcolor{KITblue}{\text{%Generate candidates + $\mathcal{L}'\leftarrow\left\{ \boldsymbol{\hat{c}}'\in\mathbb{F}_2^n: \hat{c}'_i=\hat{c}_i, i\notin \mathcal{I}' \text{ and } \hat{c}'_i\in\mathbb{F}_2, i\in \mathcal{I}' \right\} + %\left(\boldsymbol{\hat{c}}_{l}\right)_{l=1}^{2^N} + $ + %by varying bits in $\mathcal{I}$ + }} + $\vspace{1mm} + %\STATE \hspace{20mm} \textcolor{KITblue}{(list of codeword candidates)} + \STATE $\textcolor{KITblue}{\textbf{return ML\textunderscore in\textunderscore the\textunderscore list}\left( + %\left(\boldsymbol{\hat{c}}_l\right)_{1=1}^{2^N} + \mathcal{L}' + \right)}$ \end{algorithmic} \end{algorithm} +\begin{algorithm} + \caption{ML-in-the-List algorithm.} + \label{alg:ml-in-the-list} + + \begin{algorithmic} + \STATE $\mathcal{L}'_\text{valid} \leftarrow \{ \boldsymbol{\hat{c}}'\in\mathcal{L}': \boldsymbol{H}\boldsymbol{\hat{c}}'=\boldsymbol{0}\}$ (select valid codewords) + % Find valid codewords within $\mathcal{L}'$ + %under $\left(\boldsymbol{\hat{c}}_{l}\right)_{1=1}^{2^N}$ + \STATE \textbf{if} $\mathcal{L}'_\text{valid}\neq\emptyset$ \textbf{do} + %no valid codewords exist + \STATE \hspace{5mm} + \textbf{return} $\arg\max \{ \langle 1-2\boldsymbol{\hat{c}}'_l, \boldsymbol{y} \rangle : \boldsymbol{\hat{c}}'_l\in\mathcal{L}'_\text{valid}\}$ + %Compute $\langle \boldsymbol{\hat{c}}'_l, \boldsymbol{\hat{c}} \rangle$ for all variations $\boldsymbol{\hat{c}}'_l\in\mathcal{L}$ + \STATE \textbf{else} + \STATE \hspace{5mm} + \textbf{return} $\arg\max \{ \langle 1-2 \boldsymbol{\hat{c}}'_l, \boldsymbol{y} \rangle : \boldsymbol{\hat{c}}'_l\in\mathcal{L}'\}$ + %Compute $\langle \boldsymbol{\hat{c}}'_l, \boldsymbol{\hat{c}} \rangle$ for valid codewords $\boldsymbol{\hat{c}}'_l\in\mathcal{L}$ + \STATE \textbf{end if} + %\STATE \textbf{return} $\boldsymbol{\hat{c}}_l$ with highest $\langle \boldsymbol{\hat{c}}_l, \boldsymbol{\hat{c}} \rangle$ + \end{algorithmic} +\end{algorithm}% +% + + %%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Simulation Results \& Discussion} Fig. \ref{fig:results} shows the FER and BER resulting from applying proximal decoding as presented in \cite{proximal_paper} and the improved -algorithm presented here when applied to a $\left( 3,6 \right)$-regular LDPC +algorithm presented in this work, when both are applied to a $\left( 3,6 \right)$-regular LDPC code with $n=204$ and $k=102$ \cite[204.33.484]{mackay}. The parameters chosen for the simulation are $\gamma = 0.05, \omega=0.05, \eta=1.5, K=200$. @@ -711,7 +733,7 @@ as a preliminary examination showed that they provide the best results for proximal decoding as well as the improved algorithm. All points were generated by simulating at least 100 frame errors. -The number $N$ of possibly wrong components selected was selected as $8$, +The number of possibly wrong components selected was selected as $N=8$, since this provides reasonable gain without requiring an unreasonable amount of memory and computational resources. % @@ -740,8 +762,18 @@ of memory and computational resources. width=\figwidth, height=\figheight, legend pos=north east, - ylabel={BER (\lineintext{}) / FER (\lineintext{dashed})}, + ylabel={BER (\lineintext{}), FER (\lineintext{dashed})}, ] + \addplot+[FERPlot, mark=o, mark options={solid}, scol0, forget plot] + table [x=SNR, y=FER, col sep=comma, + discard if gt={SNR}{9}] + {res/bp_20433484.csv}; + + \addplot+[BERPlot, mark=*, scol0] + table [x=SNR, y=BER, col sep=comma, + discard if gt={SNR}{7.5}] + {res/bp_20433484.csv}; + \addlegendentry{BP}; \addplot+[FERPlot, mark=o, mark options={solid}, scol1, forget plot] table [x=SNR, y=FER, col sep=comma, @@ -785,12 +817,12 @@ of memory and computational resources. A noticeable improvement can be observed both in the FER as well as the BER. The gain varies significantly -with the SNR (which is to be expected, since with higher SNR values the number -of bit errors decreases, making the correction of those errors in the -``ML-in-the-list'' step more likely). +with the SNR, which is to be expected since higher SNR values result in a decreased number +of bit errors, making the correction of those errors in the +``ML-in-the-list'' step more likely. For an FER of $10^{-6}$, the gain is approximately $\SI{1}{dB}$. -Similar behavior can be observed with various other codes. -No immediate relationship between the code length and the gain was observed +Similar behavior was observed with a number of different codes, e.g., \cite[\text{PEGReg252x504, 204.55.187, 96.3.965}]{mackay}. +Furthermore, no immediate relationship between the code length and the gain was observed during our examinations. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%